In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.
For a given natural number k, a number n is called k-perfect (or k-fold perfect) iff the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect iff it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number.
It can be proven that:
- For a given prime number p, if n is p-perfect and p does not divide n, then pn is (p+1)-perfect.
- If 3n is 4k-perfect and 3 does not divide n, then n is 3k-perfect.
Smallest k-perfect numbers
The following table gives an overview of the smallest k-perfect numbers for k <= 7 (cf. Sloane's A007539):
| k | Smallest k-perfect number | Found by |
|---|---|---|
| 2 | 6 | ancient |
| 3 | 120 | ancient |
| 4 | 30240 | René Descartes, circa 1638 |
| 5 | 14182439040 | René Descartes, circa 1638 |
| 6 | 154345556085770649600 | unknown |
| 7 | 141310897947438348259849402738485523264343544818565120000 | unknown |
